Transfer-Matrix Monte Carlo Estimates of Critical Points in the Simple Cubic Ising, Planar and Heisenberg Models

TL;DR

This paper discusses an improved transfer-matrix Monte Carlo method for estimating critical points in lattice spin models, demonstrating reduced noise and accurate results for various models with efficient computation.

Contribution

It introduces a similarity transformation technique to enhance the transfer-matrix Monte Carlo algorithm, enabling more precise critical point estimates in spin models.

Findings

Reduced statistical noise via similarity transformation.

Accurate critical point estimates for 2D and 3D spin models.

Efficient computation with modest computer time.

Abstract

The principle and the efficiency of the Monte Carlo transfer-matrix algorithm are discussed. Enhancements of this algorithm are illustrated by applications to several phase transitions in lattice spin models. We demonstrate how the statistical noise can be reduced considerably by a similarity transformation of the transfer matrix using a variational estimate of its leading eigenvector, in analogy with a common practice in various quantum Monte Carlo techniques. Here we take the two-dimensional coupled $XY$-Ising model as an example. Furthermore, we calculate interface free energies of finite three-dimensional O($n$) models, for the three cases $n=1$, 2 and 3. Application of finite-size scaling to the numerical results yields estimates of the critical points of these three models. The statistical precision of the estimates is satisfactory for the modest amount of computer time spent.